First-order gradients of skew rays of axis-symmetrical optical systems

Psang Dain Lin; Chuang-Yu Tsai

doi:10.1364/JOSAA.24.000776

Journal of the Optical Society of America A
Vol. 24,
Issue 3,
pp. 776-784
(2007)
•https://doi.org/10.1364/JOSAA.24.000776

First-order gradients of skew rays of axis-symmetrical optical systems

Psang Dain Lin and Chuang-Yu Tsai

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Psang Dain Lin and Chuang-Yu Tsai, "First-order gradients of skew rays of axis-symmetrical optical systems," J. Opt. Soc. Am. A 24, 776-784 (2007)

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Abstract

Current commercial software for analysis and design of optical systems use finite difference (FD) approximation methodology to estimate the gradient matrix of a ray with respect to system variables. However, FD estimates are intrinsically inaccurate, subject to gross error when the denominator is excessively small relative to the numerator. We avoid these problems and determine these gradients by the application of Snell’s law. We give the background and basics for determining the first-order gradients of skew rays of optical systems, whereby the differential vector of any ray can be estimated by the product of the developed gradient matrix and differential changes of system variables. The most important application is for optical design by use of optimization methods where the merit function is defined as the spot size. FD used for such optimization is slow for large systems and subject to inaccuracy. The presented methodology is shown to be accurate and computationally faster than traditional FD. Two illustrative examples are provided to validate the proposed method.

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	$\partial P_{4 x} ∕ \partial β_{0}$
$Δ β_{0}$	Finite Difference	$Error %$
$1 \times 10^{- 1}$	0.033140	95.7
$5 \times 10^{- 2}$	0.024902	47.1
$1 \times 10^{- 3}$	0.017100	0.93
$1 \times 10^{- 5}$	0.016935	0.01
$1 \times 10^{- 7}$	0.016931	$- 0.01$
$1 \times 10^{- 9}$	0.016971	0.22
$1 \times 10^{- 11}$	0.015267	$- 9.8$
$1 \times 10^{- 13}$	$- 1.272221$	$- \infty$

Element j	Parameters
Element j	$t_{e j x}$	$t_{e j y}$	$t_{e j z}$	$ω_{e j x}$	$ω_{e j y}$	$ω_{e j z}$	$ξ_{2 j - 2}$	$R_{2 j - 1}$	$ξ_{2 j - 1}$	$q_{j}$	$R_{2 j}$
$j = 1$	0	$t_{e 1 y}$	0	0	0	0	1	$R_{1}$	$ξ_{1}$	$q_{1}$	$R_{2}$
$j = 2$	0	18.9	0	0	0	0	1		1	5.3
$j = 3$	0	$t_{e 3 y}$	0	0	0	0	1

	Variables (Unit, millimeter)
	$t_{e 1 y}$	$R_{1}$	$ξ_{1}$	$q_{1}$	$R_{2}$	$t_{e 3 y}$	Φ	CPU Time
Initial guess	$- 24.0$	12.0	1.4	2.1	25.1	$- 40.0$	103.67
Lower bound	$- 24.0$	12.0	1.3	1.3	25.0	$- 30.0$	121.38
Upper bound	$- 28.0$	14.0	2.0	5.1	35.0	$- 150.0$	5756.4
Solution from DBCONE	$- 28.0$	14.0	1.861	1.3	25.0	$- 78.897$	0.04910	$34 s$
Solution from DBCONG	$- 28.0$	14.0	1.859	1.3	25.0	$- 78.839$	0.04909	$4 s$

Element j	Parameters
Element j	$t_{e j x}$	$t_{e j y}$	$t_{e j z}$	$ω_{e j x}$	$ω_{e j y}$	$ω_{e j z}$	$ξ_{2 j - 2}$	$R_{2 j - 1}$	$ξ_{2 j - 1}$	$q_{j}$	$R_{2 j}$
$j = 1$	0	$t_{e 1 y}$	0	0	0	0	1	$R_{1}$	$ξ_{1}$	$q_{1}$	$R_{2}$
$j = 2$	0	$t_{e 2 y}$	0	0	0	0	1	$R_{3}$	$ξ_{3}$	$q_{3}$	$R_{4}$
$j = 3$	0	24.8	0	0	0	0	1		1	7.6
$j = 4$	0	$t_{e 4 y}$	0	0	0	0	1	$R_{7}$	$ξ_{7}$	$q_{4}$	$R_{8}$
$j = 5$	0	$t_{e 5 y}$	0	0	0	0	1	$R_{9}$	$ξ_{9}$	$q_{5}$	$R_{10}$
$j = 6$	0	$t_{e 6 y}$	0	0	0	0	1

	$\partial P_{4 x} ∕ \partial β_{0}$
$Δ β_{0}$	Finite Difference	$Error %$
$1 \times 10^{- 1}$	0.033140	95.7
$5 \times 10^{- 2}$	0.024902	47.1
$1 \times 10^{- 3}$	0.017100	0.93
$1 \times 10^{- 5}$	0.016935	0.01
$1 \times 10^{- 7}$	0.016931	$- 0.01$
$1 \times 10^{- 9}$	0.016971	0.22
$1 \times 10^{- 11}$	0.015267	$- 9.8$
$1 \times 10^{- 13}$	$- 1.272221$	$- \infty$

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Equations (47)

Journal of the Optical Society of America A